×

Iterative algorithms for hierarchical fixed points problems and variational inequalities. (English) Zbl 1205.65192

Summary: This paper deals with a method for approximating a solution of the fixed point problem: find \(\tilde{x}\in H\); \(\tilde{x}=(\mathrm{proj}_{F(t)}S)\tilde{x}\), where \(H\) is a Hilbert space, \(S\) is some nonlinear operator and \(T\) is a nonexpansive mapping on a closed convex subset \(C\) and \(\mathrm{proj}_{F(t)}\) denotes the metric projection on the set of fixed points of \(T\). First, we prove a strong convergence theorem by using a projection method which solves some variational inequality. As a special case, this projection method also solves some minimization problems. Secondly, under different restrictions on parameters, we obtain another strong convergence result which solves the above fixed point problem.

MSC:

65J15 Numerical solutions to equations with nonlinear operators
65K15 Numerical methods for variational inequalities and related problems
49J40 Variational inequalities
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Yamada, I.; Ogura, N., Hybrid steepest descent method for the variational inequality problem over the fixed point set of certain quasi-nonexpansive mappings, Numer. funct. anal. optim., 25, 619-655, (2004) · Zbl 1095.47049
[2] Stampacchia, G., Formes bilineaires coercitives sur LES ensembles convexes, C. R. acad. sci., Paris, 258, 4413-4416, (1964) · Zbl 0124.06401
[3] Aslam Noor, M.; Inayat Noor, K., Self-adaptive projection algorithms for general variational inequalities, Appl. math. comput., 151, 659-670, (2004) · Zbl 1053.65048
[4] Aslam Noor, M., Some developments in general variational inequalities, Appl. math. comput., 152, 199-277, (2004) · Zbl 1134.49304
[5] Shi, P., Equivalence of variational inequalities with wiener – hopf equations, Proc. amer. math. soc., 111, 339-346, (1991) · Zbl 0881.35049
[6] Xu, H.K.; Kim, T.H., Convergence of hybrid steepest-descent methods for variational inequalities, J. optim. theory appl., 119, 185-201, (2003) · Zbl 1045.49018
[7] Yao, Y.; Yao, J.C., On modified iterative method for nonexpansive mappings and monotone mappings, Appl. math. comput., 186, 1551-1558, (2007) · Zbl 1121.65064
[8] Censor, Y.; Iusem, A.N.; Zenios, S.A., An interior point method with Bregman functions for the variational inequality problem with paramonotone operators, Math. program., 81, 373-400, (1998) · Zbl 0919.90123
[9] Yao, J.C., Variational inequalities with generalized monotone operators, Math. oper. res., 19, 691-705, (1994) · Zbl 0813.49010
[10] Peng, J.W.; Yao, J.C., A new hybrid-extragradient method for generalized mixed equilibrium problems, fixed point problems and variational inequality problems, Taiwanese J. math., 12, 1401-1432, (2008) · Zbl 1185.47079
[11] Luo, Z.Q.; Pang, J.S.; Ralph, D., Mathematical programs with equilibrium constraints, (1996), Cambridge University Press Cambridge
[12] Cabot, A., Proximal point algorithm controlled by a slowly vanishing term: applications to hierarchical minimization, SIAM J. optim., 15, 555-572, (2005) · Zbl 1079.90098
[13] Solodov, M., An explicit descent method for bilevel convex optimization, J. convex anal., 14, 227-237, (2007) · Zbl 1145.90081
[14] Moudafi, A., Krasnoselski – mann iteration for hierarchical fixed-point problems, Inverse problems, 23, 1635-1640, (2007) · Zbl 1128.47060
[15] Byrne, C., A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse problems, 20, 103-120, (2004) · Zbl 1051.65067
[16] Censor, Y.; Motova, A.; Segal, A., Perturbed projections and subgradient projections for the multiple-sets split feasibility problem, J. math. anal. appl., 327, 1244-1256, (2007) · Zbl 1253.90211
[17] Xu, H.K., A variable krasnoselski – mann algorithm and the multiple-set split feasibility problem, Inverse problems, 22, 2021-2034, (2006) · Zbl 1126.47057
[18] Yao, Y.; Liou, Y.-C., Weak and strong convergence of krasnoselski – mann iteration for hierarchical fixed point problems, Inverse problems, 24, 1, 015015, (2008), (8 pp) · Zbl 1154.47055
[19] Mainge, P.E.; Moudafi, A., Strong convergence of an iterative method for hierarchical fixed point problems, Pac. J. optim., 3, 529-538, (2007) · Zbl 1158.47057
[20] Cianciaruso, F.; Marino, G.; Muglia, L.; Yao, Y., On a two-steps algorithm for hierarchical fixed points problems and variational inequalities, J. inequal. appl., 2009, (2009), 13 pages, Article ID 208692
[21] Chen, R.; Su, Y.; Xu, H.K., Regularization and iteration methods for a class of monotone variational inequalities, Taiwanese J. math., 13, 2B, 739-752, (2009) · Zbl 1179.58008
[22] Lu, X.; Xu, H.K.; Yin, X., Hybrid methods for a class of monotone variational inequalities, Nonlinear anal., 71, 1032-1041, (2009) · Zbl 1176.90462
[23] Xu, H.K., Viscosity method for hierarchical fixed point approach to variational inequalities, Taiwanese J. math., 14, 2, 463-478, (2010) · Zbl 1215.47099
[24] Cianciaruso, F.; Colao, V.; Muglia, L.; Xu, H.K., On an implicit hierarchical fixed point approach to variational inequalities, Bull. aust. math. soc., 80, 117-124, (2009) · Zbl 1168.49005
[25] Goebel, K.; Kirk, W.A., ()
[26] Marino, G.; Xu, H.K., A general iterative method for nonexpansive mappings in Hilbert spaces, J. math. anal. appl., 318, 1, 43-52, (2006) · Zbl 1095.47038
[27] Xu, H.K., Iterative algorithms for nonlinear operators, J. lond. math. soc., 2, 1-17, (2002)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.